On certain moment ratios of a class of discrete distributions

Two moment ratios are proposed for their uses In discriminating member among a family of dlscrete distributions and In approximating a member by another one Approximation of the Generalized Poisson( Borel–Tanner) and Neyman Type A distributions by the negatlve blnomial are also given.     

An empirical bayes approach to a variables acceptance sampling plan problem

An empirical Bayes approach to a variables acceptance sampling plan problem is presented and an empirical Bayes rule is developed which is shown to be asymptotically optimal under general conditions. The problem considered is one in which the ratio of the costs of accepting defective items and rejecting non-defective items is specified. Sampling costs are not considered and the size of the sample taken from each lot is fixed and constant. The empirical Bayes estimation of the Bayes rule is shown to require the estimation of a conditional probability. An estimator for conditional probabilities of the form needed is derived and shown to have good asymptotic properties.     

Estimation of the derivative of a density and related functions

Let Y₁,…,Y _n, (Y₁ <Y₂<…<Y _n) be the order statistics of a random sample from a distribution F with density f on the realline. This paper gives a class of estimators of the derivativef'(x) of the density f at points x for which f has

a continuoussecond derivative. These estimators are based on spacings inthe order statistics Y_j+kn -y _j j = 1,…,n-k_n,k_n<n.     

Detecting a change in variances

The likelihood ratio test for testing for a change in a sequence of variances of normal populations is derived. The alternative hypothesis considered is of a one-sided nature. For the test, the conservativeness of the Sidak bound is shown and the asymptotic version of the Sidak bound is also constructed. These bound are compared with the Bonferroni bound and the Worsley bound, using the Monte Carlo method. Finally Hsu's data of stock market returns is reanalysed, using the test.     

A test for randomness of the environments in a branching process

This paper discusses an approximate score test for testing randomness of environments in a branching process without observing the environments. Using an appropriate martingale central limit theorem the asymptotic null distribution of test statistic is shown to be normal. When the offspring distribution is Poisson, the detail derivation of asymptotic distribution of the test statistic is presented.     

Justifying an Invalid Derivation of the Probability of a Union of Events

An appealing, but invalid, derivation of the probability that at least one of n events occurs is justified, using a particular definition of subtraction of events. The probabilities that exactly m and at least m of the n events occur are derived similarly.     

Biased estimation of a variance

Summary Let , whereX _i are i.i.d. random variables with a finite variance σ² and is the usual estimate of the mean ofX _i. We consider the problem of finding optimal α with respect to the minimization of the expected value of |S ²(σ)−σ²|^k for variousk and with respect to Pitman's nearness criterion. For the Gaussian case analytical results are obtained and for some non-Gaussian cases we present Monte Carlo results regarding Pitman's criteron. This research was supported by Science Fund of Serbia, grant number 04M03, through Mathematical Institute, Belgrade.     

On Francis Galton's Difference Problem

Francis Galton proposed to split the money available for the first two prizes in a competition according to some ratio X, depending on the marks of the three best competitors, but invariant under change of location or scale of the marks. Assuming normality, Galton found that EX is about .75 and empirically he observed that X is nearly uniformly distributed between and 1. Our main purpose is to show that Galton was indeed right for a wide class of underlying distributions. As the number of competitors tends to ∞, the ratio X tends (in distribution) to a uniform random variable.     

Conditional Least Squares Estimators for the Offspring Mean in a Subcritical Branching Process with Immigration

Consider a Bienayme–Galton–Watson process with generation-dependent immigration, whose mean and variance vary regularly with non negative exponents α and β, respectively. We study the estimation problem of the offspring mean based on an observation of population sizes. We show that if β <2α, the conditional least squares estimator (CLSE) is strongly consistent. Conditions which are sufficient for the CLSE to be asymptotically normal will also be derived. The rate of convergence is faster than n ^?1/2, which is not the case in the process with stationary immigration.     

The numerical evaluation of the distribution function of a bilinear from to a quadratic from with econometric examples

This paper considers a general and computationally convenient method of evaluating the distribution function of statistics that are the ratio of a bilinear form to a quadratic form. Numerous Economemc applications of the method are given.     

The numerical evaluation of the distribution function of a bilinear from to a quadratic from with econometric examples

This paper considers a general and computationally convenient method of evaluating the distribution function of statistics that are the ratio of a bilinear form to a quadratic form. Numerous Economemc applications of the method are given.     

The bivariate linear model: a geometris approach

The geometric approach to the general linear model is in accessible to the majorityof statistics students be cause the computations require matrix algebra.This article presents the geometric approach for the special case of the bivariate linear model,for which the only tool require dis the in ner product.The geometric approach is introduced by showing the dual2-dimensional and5-dimensional representations of several bivariate samples x of size5.The assumptions of the bivariate model are stated geometrically,and the distributions of the regression coefficient sare derived.Theanalysis of variance(ANOVA)right triangle is pictured and the sides of the triang leare associated with their corresponding entries in the ANOVA table.     

Non-parametric Estimation of the Death Rate in Branching Diffusions

We consider finite systems of diffusing particles in with branching and immigration. Branching of particles occurs at position dependent rate. Under ergodicity assumptions, we estimate the position-dependent branching rate based on the observation of the particle process over a time interval [0, t ]. Asymptotics are taken as t  → ∞. We introduce a kernel-type procedure and discuss its asymptotic properties with the help of the local time for the particle configuration. We compute the minimax rate of convergence in squared-error loss over a range of Hölder classes and show that our estimator is asymptotically optimal.     

On the mvue for the distribution in the multivariate normal regression

In the multivariate normal regression setting, the estimability of a distribution is studied generalizing earlier results for the univariate case. The MVUE of an estimable distribution is obtained.     

利用概率模型估算广东篮球彩票的中奖概率

2006年足球单场竞猜异常火爆,而全国联网篮彩却一直惨淡经营,竞猜型广东篮球彩票也只试行四个月就夭折,主要问题是篮球玩法和奖金设置不够合理.如何挽救低迷中的篮彩.是一个值得研究的课题,为使广东篮彩成为未来篮彩市场的主流.本文建议提高广东篮彩一等奖的返奖率,并给予保本优惠,再增设二等奖,在此基础上利用美国全国篮球联赛2004-2005赛季的数据建立概率模型来估计中奖概率,为奖金合理设置提供有益的参考.     

Efficient sequential estimation in a markov branching process with immigration

Sequential estimation of parameters In a continuous time Markov branching process with Immigration with split rate λ₁ Immigration rate λ₂, offspring distribution {p₁j≥O) and Immigration distribution {p₂j≥l} is considered. A sequential version of the Cramér-Rao type information inequality is derived which gives a lower bound on the variances of unbiased estimators for any function of these parameters. Attaining the lower bounds depends on whether the sampling plan or stopping rule S, the estimator f, and the parametric function g = E(f) are efficient. All efficient triples (S,f,g) are characterized; It Is shown that for i = 1,2, only linear combinations of λ_ip_{ij j}'s or their ratios are efficiently estimable. Applications to a Yule process, a linear birth and death process with immigration and an M/M/∞ queue are also considered     

Factors that influence reading and comprehension of branching instructions in self–administered questionnaires

Summary: In this paper we examine the tendency for branching instructions to be ignored, misread, or otherwise not appropriately followed so that item nonresponse occurs for follow–up questions. The potential influence on branching errors of seven features of question complexity are examined, including high number of question words, high number of answer categories, last categories branch, all categories branch, write–in responses, location at the bottom of a page, and high distance between the answer box and branching instruction. A logistic regression analysis revealed that question complexity had a tendency to increase certain errors, but not others.* A more detailed version of this paper with additional analysis and discussion is available at http://www.sesrc.wsu.edu/dillman/. The opinions expressed here are those of the authors, not necessarily of the institutions where they presently work or the U.S. Census Bureau, which provided financial support for the collection of these data. We would like to thank Aref Dajani and Yves Thibaudeau for their advice on the analysis used in this paper.